OnlyState: Zero-Knowledge Prime Integer Relations

The Problem

On-chain identity and membership systems suffer from state bloat. Every member is a wallet address stored on-chain. As groups grow, gas costs explode. A DAO with 100,000 members requires storing 100,000 addresses — and every membership change requires a transaction.

OnlyState eliminates this entirely.

O(1) State Formula

An entire social group of 100,000 members is represented on-chain as exactly one 16-element array — the group's Thue-Morse sequence. The Prouhet-Tarry-Escott power-sum equations verified by the ZK circuit ensure membership without storing a single wallet address.

No databases. No identity arrays. No state bloat. Just pure Proof-of-Coherence.

Architecture

Client Layer (Browser / WASM)

Users enter a private vector. snarkjs runs the ZK circuit locally, generating a Groth16 proof (pA, pB, pC) — 300–500 bytes — and submits it to chain.

On-Chain Layer (Solidity)

• OnlyState.sol — createGroup(), proveCoherenceAndJoin(), verifyMembership()
• OSTA.sol — ERC-20 token minted exclusively by solving the ZK-PIR circuit
• Verifier.sol — auto-generated by snarkjs, verifies Groth16 proofs on-chain

Off-Chain Coherence Nodes (Rust)

Axum + tokio + redb (persistent KV) + moka (in-memory cache). Nodes index keyword hashes to encrypted IPFS CIDs and serve search results for ZK-gated decryption.

The Mathematics

Prouhet-Tarry-Escott (PTE)

The PTE problem asks: given two sets of integers, do they have equal power sums up to degree k? OnlyState uses this as a membership proof: a private vector V satisfies equal power sums against the group's public state.

Thue-Morse Sequence

The public group state is a Thue-Morse sequence — the same aperiodic binary sequence that underpins PIR. This creates a direct link between OnlyState and our PIR foundation.

Groth16 / snarkjs

The ZK proving system generates a 300-byte proof from the private vector. The proof demonstrates knowledge of a valid member vector without revealing which member. On-chain verification costs ~250k gas — fixed, regardless of group size.

BN254 Scalar Field

All ZK arithmetic happens in the BN254 prime field. No floats. Integers are scaled ×10⁶ for precision.

The OSTA Token

OSTA is minted exclusively by solving the ZK-PIR circuit and submitting a valid Groth16 proof on-chain. No staking. No time-based emissions. Only math.

Utility:

• Search fees → Coherence Nodes
• State creation burns
• Node staking
• DAO voting weight
• Compute bounties for ZK proof outsourcing

Implementation Status

• Demo playground — interactive React UI with Enclaves, Oracle, Miner, Attention Gate
• Circom ZK circuit — zk_pir.circom (N=16, C=4)
• Smart contracts — OSTA.sol + OnlyState.sol + Foundry tests
• Rust PIR Core — circuit.rs, proof.rs, verifier.rs
• Rust Coherence Node — axum + redb + moka indexer
• Browser WASM integration — snarkjs prover in demo
• Trusted setup ceremony — real MPC ceremony for mainnet
• Testnet deployment — Arbitrum Sepolia
• Cryptographic audit — ZK circuit audit (Veridise / Zellic)

Why It Matters

OnlyState demonstrates that PIR is not limited to AI governance. The same Thue-Morse equilibrium that detects market regime shifts and neural network instability can also serve as the foundation for cryptographic proofs of group membership.

This is PIR applied to blockchain: compressing infinite social graphs into a single integer, proving membership without revealing identity, and minting tokens through mathematics alone.